PHYSICAL REVIEW B 87, 035404 (2013)

Electronic structure of boron nitride sheets doped with carbon from first-principles calculations

Natalia Berseneva,1 Andris Gulans,1 Arkady V. Krasheninnikov,1,2 and Risto M. Nieminen1

1COMP Centre of Excellence and Department of Applied Physics, P.O. Box 11100, FI-00076 Aalto University, Espoo, Finland2Department of Physics, University of Helsinki, P.O. Box 43, FI-00014, Finland

(Received 15 May 2012; published 2 January 2013)

Using density functional theory with local and non-local exchange and correlation (XC) functionals, as well asthe Green’s function quasiparticle (GW) approach, we study the electronic structure of hexagonal boron nitride(h-BN) sheets, both pristine and doped with carbon. We show that the fundamental band gap in two-dimensionalh-BN is different from the gap in the bulk material, and that in the GW calculations the gap depends on theinterlayer distance (separation between the images of the BN layers within the periodic supercell approach) dueto the non-local nature of the GW approximation, so that the results must be extrapolated to infinitely largeseparations between the images. We further demonstrate by the example of carbon substitutional impuritiesthat the local and hybrid XC functionals give a qualitatively correct picture of the impurity states in the gap.Finally, we address the effects of many important parameters, such as the choice of chemical potential, and atomdisplacement cross sections for the substitutional process during electron-beam-mediated doping of h-BN sheetswith carbon atoms. Our results shed light on the electronic structure of pristine and doped h-BN and should furtherhelp to optimize the postsynthesis doping of boron nitride nanostructures stimulated by electron irradiation.

DOI: 10.1103/PhysRevB.87.035404 PACS number(s): 81.05.ue, 73.20.Hb, 73.22.−f, 61.72.−y

I. INTRODUCTION

Isolation of single sheets of graphene,1 a truly two-dimensional (2D) material, and the discovery of its uniqueproperties2 have given rise to vigorous research of other 2Dsystems.3–5 Among them, a structural analog of graphene, asingle sheet of hexagonal boron nitride (h-BN) comprisedof alternating boron and nitrogen atoms, has received par-ticular attention.6–9 Similar to graphene, h-BN has excellentmechanical properties and high thermal conductivity, while itselectronic structure is different: in comparison to semimetallicgraphene with zero band gap, 2D h-BN is a wide (more than5 eV) band gap semiconductor (see Ref. 10 for an overview ofthe properties of BN nanostructures).

Both graphene11–14 and 2D h-BN15,16 can be grown usingthe chemical vapor deposition (CVD) technique. Keeping inmind the similarities between the atomic structures of thesesystems (the mismatch between the lattice constants of thematerials is only 2%) and the radically different electronicproperties, the growth of doped BN and hybrid boron carbonnitride (BCN) materials with tunable electronic characteristicshas been attempted.17 However, although the growth of N-doped graphene has been successfully demonstrated,18,19 thehybrid BCN material has proven to be thermodynamicallyunstable with regard to segregation to graphene and h-BN,20–22

so that the growth of homogeneous structures has not beenachieved. At the same time, CVD does not provide enoughspatial control over the growth of the structures, makingimpossible the development of well-defined graphene andBN regions within a single sheet, the composite structureswhich have been predicted23–27 to have peculiar electronic andmagnetic properties.

A different approach to the development of hybrid BCNmaterials was recently demonstrated.28,29 The method wasbased on the postsynthesis substitutional carbon doping ofh-BN sheets29 and BN nanotubes28 via in situ electron-beamirradiation inside a transmission electron microscope (TEM). Itwas found that carbon atoms coming from the decomposition

of hydrocarbon molecules (intentionally added to the TEMchamber) under the 300 keV electron beam take the positionsof boron atoms; see Fig. 1(a). Using essentially the sametechnique, C impurities were also introduced into BN sheetsat electron energies of 60 keV.30

It was later theoretically shown31 that the main driving forcefor the substitution is the energetics of the system, as it costsless energy to substitute a B atom than an N atom, especiallywhen the system is becoming charged due to the electron-beamirradiation. The kinetic effects (the displacement rates forB and N atoms) could not fully explain the preferentialsubstitution of B atoms, as at an acceleration voltage of300 kV the rates are roughly the same for B and N atoms.32

Considering the possibility of creating triangular holes9 inthe h-BN sheet, shown in Fig. 1(b), it was also suggestedthat a focused electron beam could be employed to createtriangular carbon islands or straight ribbons embedded into theBN matrix, as schematically shown in Fig. 1(c), either throughbeam-mediated substitution reactions or by “filling up” theholes. The theoretical analysis31 of the substitution process wasbased on first-principles calculations of formation energies ofvarious substitutional carbon impurities in the h-BN matrix.The formation energies of defects in finite supercells werecomputed for different charge states as functions of electronchemical potentials within the framework of density functionaltheory (DFT), using local and hybrid exchange and correlation(XC) functionals. The simulation setup and its adequacy fordescribing the experimental conditions were discussed lateron,33,34 eventually giving rise to a question of how well theDFT approach describes the electronic structure of h-BN withcarbon impurities.

The main goal of this paper is to study the electronicstructure of an h-BN sheet with carbon substitutional im-purities using advanced techniques such as DFT with hybridfunctionals and Green’s function quasiparticle (GW) methods.We first carefully simulate the electronic structure of thepristine 2D h-BN material and study the convergence of the

035404-11098-0121/2013/87(3)/035404(9) ©2013 American Physical Society

BERSENEVA, GULANS, KRASHENINNIKOV, AND NIEMINEN PHYSICAL REVIEW B 87, 035404 (2013)

electron beam

hydrocarbon molecules

irradiation-mediatedsubstitution reaction

e-e- e-e-

BNC

(a)

(c)

(b)

FIG. 1. (Color online) Schematic representation of the electron-beam-mediated substitutional doping of boron nitride monolayer.Hydrocarbon molecules decomposed by the beam provide carbonatoms, which preferentially substitute boron atoms displaced by thebeam or due to a beam-induced substitution reaction. (a) Irradiation ofthe whole sample as experimentally done in Ref. 29. (b) ExperimentalTEM image of triangular holes formed in a single BN sheet due tothe electron beam, provided by Meyer et al.9 (c) Possible way ofengineering carbon islands in the BN matrix with a focused electronbeam.

results with regard to the most important parameters. We showthat due to the non-local nature of the GW approximation,the calculated fundamental band gap in 2D h-BN dependson the interlayer distance in the simulation supercell (that isthe separation between the images of the h-BN sheet withinthe periodic supercell approach), and that the results must beextrapolated to infinitely large separation between the images.We also show, that contrary to what has been assumed in manypapers, the band gaps of bulk and 2D h-BN systems differby more than 0.5 eV. We further demonstrate, by the exampleof single- and four-carbon-atom substitutional impurities, thatthe local and hybrid XC functionals give a qualitatively correctpicture of the impurity states in the gap. Finally, we addressthe effects of many important parameters such as the choiceof chemical potential, interlayer distance, displacement crosssection of electron-beam-mediated substitution process, andthe charge state of the system.

II. COMPUTATIONAL METHODS

We carried out first-principles spin-polarized plane-waveelectronic-structure calculations within density functionaltheory, as implemented in the VASP35 code. We used projector-augmented wave potentials to describe the core electronsand the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE)36 to account for the exchange-correlationenergy of interacting electrons. As the PBE approach normallyunderestimates the band gap and thus may give rise to incorrectpositions of the impurity levels, more accurate results wereobtained with the use of the hybrid Heyd-Scuseria-Ernzerhof(HSE) functional.37 To confirm the results, we carried outeven more rigorous G0W0, GW0, and GW (Refs. 38,39)calculations on top of the PBE data. Structural optimizations

of all geometries were performed by the conjugate-gradientscheme until the maximum allowed force acting on each atomwas less than 0.02 eV/A. In most of the calculations, theone-electron Kohn-Sham wave functions were expanded overa plane-wave basis with a kinetic energy cutoff of 400 eV. Wefurther studied the dependence of the results on the kineticenergy cutoff, as described in Sec. III A.

In this work, we investigate both neutral and chargedstates of the h-BN system with impurity atoms. Supercellsof different sizes (2, 18, 32, and 72) with different numbersof B and N atoms substituted with C atoms were employed.In order to prevent spurious interlayer interaction within theperiodic supercell simulation scheme, we used a vacuumspacing between adjacent 2D h-BN layers (in the directionnormal to the sheet) of 10 A. The dependence of the resultson vacuum spacing was tested, as described in Secs. III A andIII B. In the total-energy calculations, k points in the Brillouinzone (BZ) were generated by the Monkhorst-Pack schemewith a �-centered grid. The optimal number of k points used tosample the BZ was determined by a series of convergence tests.Particularly, we used 6 × 6 × 1, 4 × 4 × 1, and 6 × 6 × 3�-centered k-point meshes for computing, respectively, the3 × 3 (18 atoms), 4 × 4 (32 atoms)/6 × 6 (72 atoms), andbulk h-BN with AB stacking. Such a simulation setup has beendemonstrated to be adequate for the modeling of defects andimpurities in 2D systems with covalent bonding.32,40,41 DFTmolecular dynamics (MD) simulations of atom dynamics afterelectron impacts were carried out as in Refs. 32 and 42.

III. RESULTS AND DISCUSSION

A. Electronic structure and band gap of the pristine h-BN sheet

Although the electronic structure of bulk and 2D h-BN ma-terials has been studied at length,39,43–45 both experimentallyand theoretically, the reported results differ considerably fromeach other. Theoretical estimates of the gap for an isolatedh-BN sheet vary from 4.6 to 7.1 eV,39,45–48 depending onthe simulation method used. Experimental data also varyfrom 4.6 to 7.0 eV.49 Moreover, it was assumed in manyworks that the gap of 2D h-BN is the same as in bulk 3Dmaterial. Interestingly enough, in spite of a large number ofexperiments on the electronic properties of bulk h-BN (seeRefs. 50 and 51 and references therein), neither the direct orindirect band gap is accurately known: the values reported inthe literature range from 3.6 to 7.1 eV; see Ref. 51. Puttingaside uncertainties in the stacking order of the layers affectingthe electronic structure,46 different experimental techniques(optical/transport measurements) used to measure the gap, andstrong excitonic effects in h-BN (which should decrease thegap by nearly 1 eV),44 it is desirable to have the accuratetheoretical value for a single h-BN sheet obtained from theGW calculations. Although such calculations have alreadybeen carried out,39 the convergence of the results with regard tothe most important parameters, such as the amount of vacuumbetween the images of the sheet (within the periodic supercellapproach) and the number of empty states accounted for, hasnot been studied before.

In order to accurately assess the gap of the isolated h-BNsheet and get further insight into the electronic properties

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ELECTRONIC STRUCTURE OF BORON NITRIDE SHEETS . . . PHYSICAL REVIEW B 87, 035404 (2013)

−10−505

K Г M K

10

−10−505

K Г M K

10(a) (b)

−10−505

K Г M K

10 (c)

FIG. 2. (Color online) (a) PBE, (b) HSE, and (c) G0W0 bandstructure for pristine h-BN sheet (2-atom unit cell) with 10 A vacuumlayer thickness. The top of the valence band within the PBE is takenas zero energy. The edges of the PBE band gap are indicated by blackdashed horizontal lines. The Greek letters stand for high-symmetrypoints in the BZ of the supercell.

of BN sheets with carbon impurities, as well as to betterunderstand the accuracy of the defect formation energyestimates, we started by calculating the band structures ofthe pristine material. The band structures of the h-BN sheetwere first calculated using the PBE and HSE approaches. Theresults are presented in Fig. 2. The values of 4.56 and 5.56 eVfor the band gap of the h-BN sheet (indirect K − � gap) wereobtained within the PBE and HSE functionals, respectively.

We also calculated the quasiparticle band gap of the h-BNsheet within the framework of the GW approximation usingthe PBE data as input. We carried out not only the single-shotG0W0 simulations [Fig. 2(c)], but also studied the role ofself-consistency in the quasiparticle spectra and applied theGW0 and GW methods.38,39

The numerical reliability of existing many-body studies ofbulk and single-layer BN45,52–55 is unclear in light of recentGW calculations done by Friedrich et al. and Shih et al.56,57

They considered quasiparticle spectra of ZnO and showed thatthe calculated band gap is very sensitive to the number ofconduction bands included and the expansion of the dielectricfunction. In order to avoid these pitfalls, we carefully studiedthe convergence issues in the context of single-layer h-BN.

The GW module of the VASP package uses the plane-wavebasis in the description of the dielectric function.58 This basis islimited by the dielectric function energy cutoff q2/2 (differentfrom the plane-wave cutoff energy), where q is the absolutevalue of the largest wave vector used in the representation. Itcan be shown59 that the quasiparticle energies εQP converge as

εQP(q) = εQP(q → ∞) + A

q3+ B

q5+ O

(1

q6

), (1)

where A and B are numerical constants. A detailed discussionof Eq. (1) will be given elsewhere.59 In the present paper,we only mention that the slow convergence of quasiparticleenergies can be explained by the divergent electron-electronrepulsion.

In practical GW calculations, we use different GW cutoffenergies. Then, quasiparticle energies are fitted to the formεQP(q) ≈ εQP(q → ∞) + A

q3 + Bq5 . This procedure allows us

to extrapolate the results to the complete-basis limit. PreviousBN studies did not apply any strategies to estimate howthe dielectric function representation affects the quasiparticlespectra.

Equation (1) underlines the contribution of rapidly varyingstates. Clearly, omitting high-energy conduction states at GWcalculations leads to an additional error. Its magnitude depends

20 40 60 80 100

6.8

6.7

6.6

6.5

6.4

6.3

6.2

6.1

E gap

(eV

)

500 eV

600 eV

300 eV400 eV

Number of unoccupied bands used (%)

FIG. 3. (Color online) The band gap extrapolated to the complete-basis limit as a function of the number of unoccupied states accountedfor at different plane-wave cutoff energies calculated using G0W0

for h-BN with 10 A vacuum layer thickness. The labels stand fordifferent plane-wave cutoff energies used for the representation ofthe Kohn-Sham eigenstates.

on q: the higher the q, the more empty states are requiredto reach the desired accuracy. Thus, the q-extrapolationprocedure described above must be applied carefully.

The contribution of discarded high-energy bands is illus-trated in Fig. 3. A limited number of unoccupied states resultsin an underestimated quasiparticle band gap. For instance,using 30% of the conduction states leads to an error of 0.4 eV.Having explored numerical convergence issues, we use allavailable unoccupied states and perform the extrapolation tothe complete basis limit in further G0W0, GW0, and GWcalculations.

The obtained G0W0, GW0, and GW quasiparticle gapsfor bulk h-BN (AB stacking) are 6.07, 6.41, and 6.90 eV,respectively. The band gap increases with each level ofself-consistency, and this observation is in agreement withthe trends observed in Refs. 38 and 39.

Apart from purely numerical issues, we also consideredhow the band gap depends on the thickness of the vacuumlayer between the images of the h-BN sheet. As the separationbetween h-BN layers increases, the PBE/HSE band gapquickly saturates to the monolayer limit, as shown in Fig. 4.

0 0.02 0.04 0.06 0.08 0.1

HSE

PBE

G0W0

GW0

GW

E gap

(eV

)

Inverse interlayer distance (1/A)

4

5

6

7

8

9

FIG. 4. (Color online) The band gap as a function of the inversedistance between the sheets (vacuum spacing between the images ofthe sheets). Dots are the results of the PBE, HSE, G0W0, GW0, andGW calculations; the lines stand for extrapolations.

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BERSENEVA, GULANS, KRASHENINNIKOV, AND NIEMINEN PHYSICAL REVIEW B 87, 035404 (2013)

TABLE I. Calculated and experimental results for bulk and planar(2D) BN.

EgG0W0EgGW0

EgGW Expt. valueStructure (eV) (eV) (eV) (eV) Refs.

Bulk 6.07 6.41 6.90 3.6–7.1 50, 512D 7.40 7.83 8.43 4.6–7.0 49

The situation is different in the GW calculations. The G0W0,GW0, and GW band gaps for an isolated h-BN sheet (at 10 Avacuum separation) are 6.60, 7.00, and 7.60 eV, respectively.The G0W0 value agrees well with the results reported inRef. 47. However, it is evident from Fig. 4 that the band gapscalculated at 40 A of the interlayer separation in h-BN aredifferent from the values calculated at 10 A and still differfrom the limit of infinite separation. This behavior contrastswith the PBE and HSE data, where a separation of 15 A issufficient to obtain a converged band gap.

Such a difference between PBE/HSE and all three quasipar-ticle methods can be explained by non-local screening effects,which are described with the GW formalism, but cannot becaptured with PBE and HSE. For a detailed discussion of thephysics behind this effect, we refer to Refs. 60 and 61. Inthe limit of large separations d between h-BN layers, the GWband gap converges as 1/d, and we use this dependence forextrapolation. Truncation of the Coulomb interaction in thedirection perpendicular to the sheets may also be a solution tothis problem.

The G0W0, GW0, and GW band gaps for an isolated h-BN sheet (at infinite separation) are 7.40, 7.83, and 8.43 eV,respectively. It is obvious that the band gaps for bulk andmonolayer h-BN differ by more than 0.5 eV (see also Table I).

B. Electronic structure and band gap of the h-BN sheet withcarbon impurities

Having analyzed the electronic structure of the pristine h-BN sheet, we calculated the band structures of doped BNsheets with triangular-shaped substitutional carbon defects.31

Examples of investigated configurations are shown in Fig. 5.Throughout this paper, we use the following notations: CB

means that a single B atom was substituted with a C atom, while4C3B1N stands for a triangular island formed by substitutingthree B and one N atoms with four C atoms, etc.

C1B3N

CB

CN

C3B1N

CNB

FIG. 5. (Color online) Examples of several structures studied inour work, including single carbon atom substitutional impurities (CB

and CN) in the h-BN sheet and four carbon atom islands with boron(4C1B3N-structure) and nitrogen (4C3B1N-structure) termination.

C6B10N

C10B6N

-5-4-3

-2

-1 0

1

2

K Г M K-5-4

-3

-2

-1 0

1

2

K Г M K

-4

-3

-2

-1

0

1

2

K Г M K-4

-3

-2

-1

0

1

2

K Г M K

spin upC10B6N

C10B6Nspin dn

C6B10Nspin up

C6B10Nspin dn

CNB

FIG. 6. (Color online) Examples of the structures with largercarbon triangular substitutional impurities in the h-BN sheet withboron (16C6B10N) and nitrogen (16C10B6N) termination together withcalculated PBE-based band structures (spin-up and spin-down resultsare presented). The Greek letters stand for high-symmetry points inthe BZ of the supercell.

In Fig. 7, we present PBE and HSE band structures for32-atom supercells with substitutional defects composed fromone to four carbon atoms (CB, CN, 4C3B1N, 4C1B3N) insertedinto the h-BN host matrix. The PBE and HSE pictures arequalitatively the same, which explains the good agreementbetween the HSE and PBE formation energies of the defectsreported earlier.31 New states associated with defects appearin the gap, either deep states close to the middle of the gapor shallow states close to the band edges. It is obvious thatthe larger the defect size, the more new states appear in theband gap (Fig. 6). Defects with the preferential substitution ofB atoms (e.g., 4C3B1N) give rise to occupied states (for zeroelectron chemical potential), while those with the preferentialsubstitution of N atoms (e.g., 4C1B3N structure) give rise toempty states.

We also carried out quasiparticle G0W0 calculations forsingle-atom impurities. The band structures for a 18-atomsystem (at 10 A interlayer separation) with CB and CN defectsare shown in Fig. 8. The positions of the impurity levelsqualitatively agree well for PBE, HSE, and G0W0 calculations.In addition, we considered the band structure dependence onthe thickness of the vacuum layer between the images ofthe h-BN sheets. Due to the high computational cost, onlycalculations with 12.5 and 15 A vacuum layer separation werecarried out for an 18-atom system with CB and CN defects.We conclude that with increasing the interlayer distance, theimpurity level moves up in energy (towards the conductionband edge).

C. Defect formation energies: The role of chemical potential

The substitutional carbon defects can appear in variouscharge states q. The formation energy Ef of a defectcomposed of KC carbon atoms inserted to replace KB

boron and KN nitrogen atoms (KB + KN = KC) in the h-BN

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ELECTRONIC STRUCTURE OF BORON NITRIDE SHEETS . . . PHYSICAL REVIEW B 87, 035404 (2013)

PBE

K M K

spin up32 BN CB

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

K M K

spin up32 BN CB

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

K M K

spin dn32 BN CB

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

K M K

spin dn32 BN CB

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

K M K

spin up32 BN CN

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

K M K

spin up32 BN CN

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

K M K

spin dn32 BN CN

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

K M K

spin up32 BN 4C1B3N

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

K M K

spin up32 BN 4C1B3N

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

K M K

spin dn32 BN 4C1B3N

Г

0

2

4

7

PBE

1

3

5

6

K M K

spin up

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

32 BN 4C3B1N

K M K

spin up

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

32 BN 4C3B1N

K M K

spin dn

Г-3

-2

0

2

4

7

-1

1

3

5

6

32 BN 4C3B1N

K M K

spin dn

Г-3

-2

0

2

4

7

HSE

-1

1

3

5

6

32 BN 4C3B1N

K M K

spin dn32 BN 4C1B3N

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

K M K

spin dn32 BN CN

Г-3

-2

0

2

4

7

PBE

-1

1

3

5

6

-3

-2

-1

FIG. 7. (Color online) Band structures of 32-atom supercells with substitutional defects composed from one to four carbon atoms (CB, CN,4C3B1N, 4C1B3N defects) inserted into the h-BN host matrix. Calculations are done involving PBE (blue lines) and HSE (red lines) approaches.Green dotted lines emphasize the impurity levels. Spin-up and spin-down results are presented. The top of the valence band within the PBE istaken as zero energy. The edges of the PBE band gap are indicated by black dashed horizontal lines. The Greek letters stand for high-symmetrypoints in the BZ of the supercell.

matrix is

Ef

(K

q

C

) = Etot(K

q

C

) − Etot(BN) + KBμB

+KNμN − KCμC + q(μe − EV ), (2)

where Etot(Kq

C) and Etot(BN) are the calculated total energiesof defected and defect-free BN systems, respectively. q is thecharge state of the defect complex, EV is the energy positionof the valence band maximum, and μe is the electron chemicalpotential (defined within the band gap with respect to the EV ).μB, μN, and μC are the chemical potentials of the elementspresent in the system. In equilibrium,

μBN = μB + μN, (3)

where μBN is the total energy per BN pair in a pristine h-BNsheet. μC is chosen as the energy per atom in a C monolayer(graphene sheet). The choice of μB and μN is defined by thegrowth conditions of BN sheets, such as stoichiometry betweenthe components.

A previous work31 showed that under N-rich conditions,the substitution of B atoms with C atoms costs less than thesubstitution of N atoms. For the neutral charge states, thedifference in energy is about 2 eV in favor of B for single-atomdefects. When a single B (N) atom is substituted with a Catom, one extra electron is added (removed) and the dominantcharge states of the CB and CN defects are +1, 0, and −1. Thelarger defect islands can support higher charge states. Positive

charging further lowers the formation energy of CB defects,while negative charging favors CN.

There are many factors that may affect the growth processduring material manufacture. Therefore, defining the properparameters and environmental conditions is crucial. Thecorrect choice of the environmental conditions is also veryimportant for electron-beam-mediated postsynthesis doping.Even though an N-rich environment better matches thedescribed experimental situation, we also investigated howthe results depend on the choice of boron chemical potentialby the example of single substitutional impurities (C atom inB- and N-atom positions). We varied μB up to the value forbulk material (B-rich environment). The calculated Ef of CB

and CN defects for the neutral charge states are presented inFig. 9 as functions of μB. The horizontal lines are formationenergies for a single carbon impurity under N-rich conditions.The lines that indicate the Ef behaviors of CB and CN ina B-rich environment intersect (Fig. 9). This crossing pointseparates the region where the substitution of B (left fromthe point) is preferable over N substitution. Contrarily, thesubstitution of N atoms becomes preferable at μB growing(right from the point) towards the energy of an isolated Batom. The value of μB = −7.62 eV corresponds to the lowestenergy α-rhombohedral phase of boron. It is computed as asum of the energy of an isolated atom in our model and thecohesive energy of the α phase.62 It should be emphasizedthat the selected potential of the μB is close to the crossing

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BERSENEVA, GULANS, KRASHENINNIKOV, AND NIEMINEN PHYSICAL REVIEW B 87, 035404 (2013)

spin up18 BN CB

-4

-2

0

2

4

6

8

K M K

spin up18 BN CB

Г-4

-2

0

2

4

6

8

PBE G0W0

spin dn

K M KГ

18 BN CB

-4

-2

0

2

4

6

8

PBEspin dn

K M KГ

18 BN CB

-4

-2

0

2

4

6

8

G0W0

spin up

K M KГ

18 BN CN

-4

-2

0

2

4

6

8

G0W0spin up

K M KГ

18 BN CN

-4

-2

0

2

4

6

8

PBEspin dn

K M KГ

18 BN CN

-4

-2

0

2

4

6

8

PBEspin dn

K M KГ

18 BN CN

-4

-2

0

2

4

6

8

G0W0

K M KГ

FIG. 8. (Color online) Electronic band structure of 18-atom BNsupercell (at 10 A vacuum separation) where a B or N atom issubstituted with a C atom. Calculations are done involving PBE (bluelines) and G0W0 (red lines) approaches. Green dotted lines emphasizethe impurity levels. Spin-up and spin-down results are presented. Thetop of the valence band within the PBE is taken as zero energy. Theedges of the PBE band gap are indicated by black dashed horizontallines. The Greek letters stand for high-symmetry points in the BZ ofthe supercell.

point. Taking into account the drop in Ef corresponding toB substitution at positive charge states,31 C atoms may stillpredominately take the positions of B atoms even in the B-richenvironment.

We also looked at the substitution process from a differentperspective. Assuming that carbon adatoms appear on theh-BN sheet due to electron-beam-mediated dissociation of

-10 -9 -8 -7 -6 -5012

3

456

E f(e

V)

μB (eV)

CN, N env.

CB, B env.

CN, B env.

CB, N env.

μΒ+μΝ = μΒΝ

FIG. 9. (Color online) Energy required to substitute B and Natoms with C for the neutral charge states as functions of μB.The horizontal lines stand for the Ef of CB and CN configurationscalculated in the N-rich environment. The vertical dotted linecorresponds to the condition defined by Eq. (3).

+2.29 eV

+1.06 eV

+2.83 eV

q=+1

+4.43 eV

+2.02 eV

q= 1

BC

NCN + N adatom

-

q=0+3.18 eV

CB + B adatom

CN + N adatom

CN + N adatom

CB + B adatom

CB + B adatom

FIG. 10. (Color online) Difference in the total energy of thesystem for the neutral and charged states of the h-BN sheet withdefects. C adatom on pristine h-BN sheet (reference system); C atomsin substitutional positions and the corresponding B or N adatom onthe h-BN sheet are investigated. Numbers stand for the total-energydifference between the reference system and a configuration shownon the right side from the arrow. q stands for the charge. It is evidentthat positive charge decreases the formation energy of substitutionalC impurity in the position of the B atom.

hydrocarbon molecules, we explored several neutral andcharged systems with adatoms: with C adatom on the pristineBN sheet, and with C atoms in substitutional positions and thecorresponding B or N adatom on the h-BN sheet (Fig. 10). Thetotal-energy analysis of those systems (Fig. 10) demonstratesthe same trend that was observed earlier in our work31—positive charging reduces the energy required to exchangea B atom with a C atom and makes it lower than the energyrequired to exchange an N atom with a C atom, while negativecharge leads to the opposite situation. The lower formationenergies of defects with N-terminating edges are consistentwith the preferential substitution of B atoms reported in theexperiments.28,29

D. Displacement energies and rates

The complete understanding of the substitution processunder the electron beam is not possible without the mi-croscopic knowledge of the dynamical effects. To estimatethe role of kinetics in the process of carbon substitution,molecular dynamics simulations of electron impacts onto theh-BN monolayer were carried out. Since we are interested inthe growth of carbon islands under electron-beam irradiation,displacement thresholds of C substitutional impurity (in theB- and N-atom position) as well as its nearest-neighbor B andN atoms were calculated. The displacement threshold (Td )is the minimum kinetic energy that should be delivered tothe atom to remove it from its position without immediaterecombination. The corresponding electron threshold energycan be estimated32,63 when Td is known. In addition, when thedisplacement threshold is known, the estimation of the dis-placement rate (σ ) can be done using the McKinley-Feshbach

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ELECTRONIC STRUCTURE OF BORON NITRIDE SHEETS . . . PHYSICAL REVIEW B 87, 035404 (2013)

40 80 120 160 200 240 280 320 360

V (keV)

2

4

6

8

10

12

14

0

σ (1

0-4/s

)

CCNCB

BCNNCB

NB

N(CB)NN

B

CB

N

CN

B(CN)

FIG. 11. (Color online) Displacement rates for B and N atoms inpristine systems, C atom in B and N position, nearest to the singleC substitutional defect B and N atoms as functions of accelerationvoltage (electron-beam energy). Black vertical dashed line indicatesthe value of the electron-beam energy used in the experiments(300 keV). All values were calculated at electron-beam current of10.0 A/cm2.

formalism for Coulomb scattering of relativistic electrons.64

In the simulations, the h-BN sheet is oriented perpendicularlyto the incident electron direction and thus maximum energytransfer from the incident electron to the atom is considered.Kinetic energy was instantaneously assigned to the recoil atom,and DFT molecular dynamics was used to investigate if theamount of energy was sufficient to take the atom away fromthe system. We neglected the effects of temperature65,66 on thedisplacement thresholds, as they give rise to a tail just belowthe threshold only and do not change the shapes of the curvesat higher voltages.

We found that Td for the C atom (independent of the B or Natom substitution) is 1.5 eV less than for a B atom in the pristinesystem, giving a rise to a 30% increase in the displacement rate(for 300 keV used in the simulations) as compared to that fora B atom in the pristine material (Fig. 11). This means thatthe substitution of individual atoms can hardly be explainedby dynamical effects only. The displacement threshold for thenearest B-neighbor atoms (CN defect) drops by 1.2 eV, and by3.6 eV for neighboring N atoms (CB defect) as compared to thecorresponding values in the pristine system. The knowledge ofthe displacement rates for different types of atoms, along withthe information on the energetics of different configurations

and partial pressure of the volatile components in the chamber,should make it possible to use the Monte Carlo methods tomodel the kinetics of the process.

IV. CONCLUSIONS

To conclude, using DFT approaches with local and non-local XC functionals and the GW method, we calculatedthe electronic structure of h-BN sheets. We considered bothpristine systems and those doped with carbon. We showed thatthe fundamental band gap in 2D h-BN is different from thegap in the bulk material. We also demonstrated that due to thenon-local nature of the GW approximation, the gap calculatedusing the GW approach depends on the interlayer separationbetween the images of the layers within the periodic supercellapproach, so that the results corresponding to the isolated 2Dsystem must be extrapolated to the limit of infinitely largeseparations between the images. We further showed by theexample of carbon substitutional impurities that the local andhybrid XC functionals give a qualitatively correct picture ofthe impurity states in the gap. Finally, we address the effectsof many important parameters such as the choice of chemicalpotential, and estimated atom displacement cross sections forthe substitutional process during the electron-beam-mediateddoping of h-BN sheets with carbon atoms. Our results shedlight on the electronic structure of pristine and doped h-BN and should further help to optimize the postsynthesisdoping of boron nitride nanostructures stimulated by theelectron irradiation. Specifically, by using a focused electronbeam and simultaneously charging the sample, one canmanufacture spatially localized hybrid structures which shouldexhibit intriguing electronic and magnetic properties.67,68 Thetechnique, which can be referred to as a combination ofelectron irradiation and beam-assisted deposition, can beapplied to various 2D materials, including recently reportednon-organic 2D structures.4 Indeed, filling of vacancies createdby the electron beam in molybdenum disulfide was recentlydemonstrated in the experiments.69

ACKNOWLEDGMENTS

This work was supported by the Finnish Academy ofScience and Letters, and the Academy of Finland through theCentre of Excellence and other projects. We also acknowledgethe Finnish IT Center for Science (CSC) for generous grantsof computer time.

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